I'm on a Vector Analysis class and was given a problem on the convergence/divergence of sequences on $\mathbb{R}^n$. The definition that we're using states that:
A sequence $x_n$ $\in \mathbb{R}^n$ converges to $a \in \mathbb{R}^n$ if for every $\color{red}{\text{neighborhood }} U$ for point $a$ there exists $k_0 \in \mathbb{N}$ such that $x_k \in U$ for every $k \geqslant k_0.$
I'm a bit confused about the definition here. By neighbourhood do they mean something like the epsilon-neighborhood that's more related to topology than vector analysis?
Also the notation was described as $$\lim_{k\to\infty} ||x_k-a||=0$$
I guess the $ ||x_k-a||$ here is not an absolute value, but the Euclidean norm?